Riemann's theorem for quantum tilted rotors

a r X i v :n u c l -t h /9210012v 1 19 O c t 1992Riemann’s theorem for quantum tilted rotorsG.Rosensteel and A.L.GoodmanDepartment of Physics,Tulane University,New Orleans,Louisiana 70118The angular momentum,angular velocity,Kelvin circulation,and vortex velocity vectors of a quantum Riemann rotor are proven to be either (1)aligned with a principal axis or (2)lie in a principal plane of the inertia ellipsoid.In the second case,the ratios of the components of the Kelvin circulation to the corresponding components of the angular momentum,and the ratios of the components of the angular velocity to those of the vortex velocity are analytic functions of the axes lengths.PACS:21.60.Ev,21.60.FwTypeset Using REVTEXA classical Riemann rotor is a uniform density fluid with an ellipsoidal boundary and a velocity field that is a linear function of position.Riemann fluids model rotating stars and galaxies,[1,2]spinning gas clouds,[3]and rotating nuclei.[4,5]Since linear velocityfields spanthe dynamical continuum from rigid rotation to irrotational flow,the Riemann model is sufficiently general to model most collective rotational systems.The vector observablesthat measure the character of the rotation are the angular momentum Land the Kelvin circulation C ;[5–8]the vector variables conjugate to L and C are the angular velocity ωand the vortex velocity λ.Tilted nuclear rotors for which ωis not aligned with a principal axis are a topic of continuing interest.[9–17,?,19,20]Recently,a quantum theory of Riemann rotors was formulated by simultaneous angular and vortex cranking of the nuclear mean field Hamiltonian,H ω λ=H 0−ω· L + λ· C,(1)where the mean field is approximated by the anistropic oscillator potential,H 0=−¯h 22m (ω2x x 2+ω2y y 2+ω2z z 2).(2)Inglis’s cranking formula determines the collective energy of an A -nucleon system,[21,22]T ( ω, λ)= ph | p | ω· L − λ· C|h |243 k =1[(a 2i +a 2j )(ω2k +λ2k )−4a i a j ωk λk ],(5)where i,j,k are cyclic,a1,a2,a3are the axes lengths of the inertia ellipsoid in units of R, and the moment of inertia of a sphere of radius R and total mass M=mA is I0=(2/5)MR2. If the nuclear volume is set equal to the spherical value4πR3/3,then the product of the dimensionless axes lengths equals unity,a1a2a3=1.Note that if the vortex velocity vanishes, then the self-consistent collective energy equals the rigid body energy,a well-known result.[24]In addition,the quantum expectations of the angular momentum and Kelvin circulation equal their classical Riemann rotor values,[1]I0L k =2 2a i a jωk−(a2i+a2j)λk .(6) These expectations are given by derivatives of the kinetic energy with respect to the angular velocity and the vortex velocity[25]L k = ∂T∂λk .(7) The collective energy may be expressed as1T( ω, λ)=˜E( ω, λ+δ λ)−˜E( ω, λ)=ǫ(ˆn× λ)· C +O(ǫ2)(10)=ǫˆn·( λ× C )+O(ǫ2)(11)Hence,at equilibrium,ˆn·( λ× C )=0for all directionsˆn,or λ× C =0.Riemann[26]proved in1860that the classical rotors in equilibrium fall into three classes:1.Rigid rotors(λ=0)which encompasses the Maclaurin spheroids and the Jacobi triaxialellipsoids,2.S-type ellipsoids for which the directions of ωand λare aligned with a principal axis,and3.Tilted ellipsoids for which the directions of ωand λlie in a principal plane.The case where ωand/or λdo not lie in a principal plane is specifically excluded.It is remarkable that Riemann’s theorem is also true for quantum cranked Riemann rotors.To prove the theorem,substitute the explicit formulae for the expectations of the angular mo-mentum and Kelvin circulation,Eq.(6),into the parallelism conditions,Eqs.(9a,9b).If the angular velocity vector ωis neither aligned with a principal axis nor lies in a principal plane,thenω1,ω2,ω3are all nonzero and the parallelism conditions constitute a set of six simultaneous equations in the three unknown ratios,λ1/ω1,λ2/ω2,λ3/ω3.It can be shown that only four of these equations are independent.Therefore,this simultaneous system is overdetermined for the three unknown ratios.Since the assumption that all three compo-nentsωk are nonzero implies a contradiction,one of the angular velocity components must vanish,sayω1.But,writing out the y-component of Eq.(9a),0=ω3 L1 =−2a2a3ω3λ1, one concludes thatλ1must also vanish.Hence, ωand λlie in a principal plane of the inertia ellipsoid.Ifω2andω3are both nonzero,then the parallelism conditions produce two independent equations in two unknown ratios whose solution isλ2/ω2=(4a21−a22+a23±q)/4a1a3λ3/ω3=(4a21+a22−a23±q)/4a1a2,(12)whereq2=(4a21−(a3+a2)2)(4a21−(a3−a2)2).(13)If only one of the components of ωis nonzero,then the angular velocity,vortex velocity, angular momentum,and Kelvin circulation vectors are all aligned with a single principal axis.The ratioλ/ωis undetermined for such an S-type ellipsoid.This completes the proof of Riemann’s theorem.Since q cannot be imaginary,there are only three types of tilted rotors.Choosing an ordering of the axes lengths in the principal plane,say a3≥a2,yields the following:•Type I.2a1≥a3+a2•Type II.2a1≤a3−a2and a2≤a1•Type III.2a1≤a3−a2and a1≤a2.Type II and Type III tilted rotors are ultradeformed prolate-like solutions for which the ratio of the longest to the shortest axis is at least three to one.Type I solutions are tilted oblate-like rotors;these have been studied in the classical macroscopic approximation where the potential energy is a sum of the attractive surface energy plus the repulsive Coulomb potential.[5]REFERENCES1S.Chandrasekhar,Ellipsoidal Figures of Equilibrium,(Yale University Press,New Haven,1969).2Norman R.Lebovitz,Ann.Rev.Astr.Astrophys.5,465(1967).3 F.J.Dyson,J.Math.Mech.18,91(1968).4R.Y.Cusson,Nucl.Phys.A114,289(1968).5G.Rosensteel,Ann.Phys.(N.Y.)186,230(1988).6O.L.Weaver,R.Y.Cusson,and L.C.Biedenharn,Ann.Phys.(N.Y.)102,493(1976). 7P.Gulshani and D.J.Rowe,Can.J.Phys.54,970(1976).8G.Rosensteel and D.J.Rowe,Ann.Phys.(N.Y.)96,1(1976).9H.Frisk and R.Bengtsson,Phys.Lett.B196,14(1987).10Alan L.Goodman,Phys.Rev.C45,1649(1992);Nucl.Phys.(to be published).11W.Nazarewicz and Z.Szymanski,Phys.Rev.C45,2771(1992).12 D.J.Thouless and J.G.Valatin,Nucl.Phys.31,211(1962).13 E.R.Marshalek,Nucl.Phys.A331,429(1979);in Proceedings of the Symposium onContemporary Physics,Philadelphia,1991(unpublished).14I.N.Mikhailov and D.Janssen,Phys.Lett.B72,303(1978);D.Janssen and I.N.Mikhailov,Nucl.Phys.A318,390(1979).15M.G.Vassanji and M.Harvey,Nucl.Phys.A344,61(1980).16 A.K.Kerman and N.Onishi,Nucl.Phys.A361,179(1981).17 F.Cuypers,Nucl.Phys.A468,237(1987).18 A.Klein and M.G.Vassanji,Phys.Rev.Lett.42,436(1979).19 F.D¨o nau,Nucl.Phys.A517,125(1990).20S.Frauendorf and T.Bengtsson,in Proceedings of the Workshop on Nuclear Structure in the Era of New Spectroscopy,Part B,Copenhagen,1989(unpublished).21 D.R.Inglis,Phys.Rev.96,1059(1954).22Peter Ring and Peter Schuck,The Nuclear Many-Body Problem,(Springer-Verlag,New York,1980).23G.Rosensteel,Phys.Rev.C,in press.24Aage Bohr and Ben R.Mottelson,Nuclear Structure,(Benjamin,Reading,1975)Vol.II.25R.P.Feynman,Phys.Rev.56,340(1939).26 B.Riemann,Abh.K.Ges.Wis.G¨o ttingen9,3(1860).。